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2007 Exceptional Regions and Associated Exceptional Hyperbolic 3-Manifolds, with an appendix by Alan W. Reid
Abhijit Champanerkar, Jacob Lewis, Max Lipyanskiy, Scott Meltzer
Experiment. Math. 16(1): 106-118 (2007).

Abstract

A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff, and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated with one of the families. It is conjectured that there are exactly six exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea census of closed hyperbolic 3-manifolds, associated with five other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated with the seventh family, we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields.

Citation

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Abhijit Champanerkar. Jacob Lewis. Max Lipyanskiy. Scott Meltzer. "Exceptional Regions and Associated Exceptional Hyperbolic 3-Manifolds, with an appendix by Alan W. Reid." Experiment. Math. 16 (1) 106 - 118, 2007.

Information

Published: 2007
First available in Project Euclid: 5 April 2007

zbMATH: 1146.57027

Subjects:
Primary: 57M50
Secondary: 57N10

Keywords: Exceptional regions , hyperbolic 3-manifolds , invariant trace field , quasi-relator

Rights: Copyright © 2007 A K Peters, Ltd.

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Vol.16 • No. 1 • 2007
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